List of functions
This page lists various googological functions arranged by growth rate according to the fast-growing hierarchy. *\(\approx\) means that two functions have comparable growth rates. *\(>\) means that one function significantly overgrows the other. *\(\geq\) means that it is not known exactly whether one function overgrows the other or not. *(limit) means that the function has many arguments, and the growth rate is found by diagonalizing over them. From \(f_0(n)\) to \(f_\omega(n)\) These functions are primitive recursive. *Addition \(a+b > f_0(n)\) *Multiplication \(a*b > f_1(n)\) *Exponentiation \(a^b \approx f_2(n)\) *Factorial (and most of its extensions) \(n! \approx f_2(n)\) *Double exponential functions \(\approx a^{a^b} \approx f_2(f_2(n)) \) *Exponential factorial \(\approx f_3(n)\) *Tetration \({^{b}a} \approx f_3(n)\) *Pentation \(a \uparrow\uparrow\uparrow b \approx f_4(n)\) *Circle notation \(\approx f_4(n)\) *Hexation \(a \uparrow^{4} b \approx f_5(n)\) *Heptation \(a \uparrow^{5} b \approx f_6(n)\) *Octation \(a \uparrow^{6} b \approx f_7(n)\) *Enneation \(a \uparrow^{7} b \approx f_8(n)\) *Decation \(a \uparrow^{8} b \approx f_9(n)\) *Undecation \(a \uparrow^{9} b \approx f_{10}(n)\) *Doedecation \(a \uparrow^{10} b \approx f_{11}(n)\) *Tredecation \(a \uparrow^{11} b \approx f_{12}(n)\) From \(f_\omega(n)\) to \(f_{\omega^\omega}(n)\) *Weak goodstein function \(g(n) \approx f_\omega(n)\) *Ackermann function \(A(n,n) \approx f_\omega(n)\) *Ackermann numbers \(\approx f_\omega(n)\), the limit of the hyper operators in general *Steinhaus-Moser notation \(\approx f_\omega(n)\) *Arrow notation (both variants) \(a \uparrow^{c} b \approx f_\omega(n)\) * Psi Notation \(\approx f_\omega(n)\) *Hyper-E notation \(E\# \approx f_\omega(n)\) (limit) *Graham's function \(g_n \approx f_{\omega+1}(n)\) *Exploding Tree Function \(E(n) \approx f_{\omega+1}(n)\) *Expansion \(a \{\{1\}\} b \approx f_{\omega+1}(n)\) *Multiexpansion \(a \{\{2\}\} b \approx f_{\omega+2}(n)\) *Powerexpansion \(a \{\{3\}\} b \approx f_{\omega+3}(n)\) *Expandotetration \(a \{\{4\}\} b \approx f_{\omega+4}(n)\) *Explosion \(a \{\{\{1\}\}\} b \approx f_{\omega 2+1}(n)\) *Multiexplosion \(a \{\{\{2\}\}\} b \approx f_{\omega 2+2}(n)\) *Powerexplosion \(a \{\{\{3\}\}\} b \approx f_{\omega 2+3}(n)\) *Explodotetration \(a \{\{\{4\}\}\} b \approx f_{\omega 2+4}(n)\) *Detonation \(\{a,b,1,4\} \approx f_{\omega 3}(n)\) *Pentonation \(\{a,b,1,5\} \approx f_{\omega 4}(n)\) *Hexonation \(\{a,b,1,6\} \approx f_{\omega 5}(n)\) *Heptonation \(\{a,b,1,7\} \approx f_{\omega 6}(n)\) *Octonation \(\{a,b,1,8\} \approx f_{\omega 7}(n)\) *Ennonation \(\{a,b,1,9\} \approx f_{\omega 8}(n)\) *Deconation \(\{a,b,1,10\} \approx f_{\omega 9}(n)\) *CG function \(cg(n) \approx f_{\omega^2}(n)\) *Megotion \(\{a,b,1,1,2\} \approx f_{\omega^2+1}(n)\) *BOX_M̃ function \(\widetilde{M}_n \approx f_{\omega^2+1}(n)\) *Multimegotion \(\{a,b,2,1,2\} \approx f_{\omega^2+2}(n)\) *Powermegotion \(\{a,b,3,1,2\} \approx f_{\omega^2+3}(n)\) *Megotetration \(\{a,b,4,1,2\} \approx f_{\omega^2+4}(n)\) *Megoexpansion \(\{a,b,1,2,2\} \approx f_{\omega^2+\omega+1}(n)\) *Multimegoexpansion \(\{a,b,2,2,2\} \approx f_{\omega^2+\omega+2}(n)\) *Powermegoexpansion \(\{a,b,3,2,2\} \approx f_{\omega^2+\omega+3}(n)\) *Megoexpandotetration \(\{a,b,4,2,2\} \approx f_{\omega^2+\omega+4}(n)\) *Megoexplosion \(\{a,b,1,3,2\} \approx f_{\omega^2+\omega 2}(n)\) *Megodetonation \(\{a,b,1,4,2\} \approx f_{\omega^2+\omega 3}(n)\) *Gigotion \(\{a,b,1,1,3\} \approx f_{(\omega^2) 2+1}(n)\) *Gigoexpansion \(\{a,b,1,2,3\} \approx f_{(\omega^2) 2+\omega}(n)\) *Gigoexplosion \(\{a,b,1,3,3\} \approx f_{(\omega^2) 2+\omega 2}(n)\) *Gigodetonation \(\{a,b,1,4,3\} \approx f_{(\omega^2) 2+\omega 3}(n)\) *Terotion \(\{a,b,1,1,4\} \approx f_{(\omega^2) 3+1}(n)\) *Petotion \(\{a,b,1,1,5\} \approx f_{(\omega^2) 4+1}(n)\) *Hatotion \(\{a,b,1,1,6\} \approx f_{(\omega^2) 5+1}(n)\) *Hepotion \(\{a,b,1,1,7\} \approx f_{(\omega^2) 6+1}(n)\) *Ocotion \(\{a,b,1,1,8\} \approx f_{(\omega^2) 7+1}(n)\) *Nanotion \(\{a,b,1,1,9\} \approx f_{(\omega^2) 8+1}(n)\) *Uzotion \(\{a,b,1,1,10\} \approx f_{(\omega^2) 9+1}(n)\) *Uuotion \(\{a,b,1,1,11\} \approx f_{(\omega^2) 10+1}(n)\) *Udotion \(\{a,b,1,1,12\} \approx f_{(\omega^2) 11+1}(n)\) *Utotion \(\{a,b,1,1,13\} \approx f_{(\omega^2) 12+1}(n)\) *Ueotion \(\{a,b,1,1,14\} \approx f_{(\omega^2) 13+1}(n)\) *Upotion \(\{a,b,1,1,15\} \approx f_{(\omega^2) 14+1}(n)\) *Uhotion \(\{a,b,1,1,16\} \approx f_{(\omega^2) 15+1}(n)\) *Uaotion \(\{a,b,1,1,17\} \approx f_{(\omega^2) 16+1}(n)\) *Uootion \(\{a,b,1,1,18\} \approx f_{(\omega^2) 17+1}(n)\) *Unotion \(\{a,b,1,1,19\} \approx f_{(\omega^2) 18+1}(n)\) *Dzotion \(\{a,b,1,1,20\} \approx f_{(\omega^2) 19+1}(n)\) *Tzotion \(\{a,b,1,1,30\} \approx f_{(\omega^2) 29+1}(n)\) *Uzzotion \(\{a,b,1,1,100\} \approx f_{(\omega^2) 99+1}(n)\) *Hurford's C function \(C(n) \approx f_{\omega^3 + \omega}(n)\) From \(f_{\omega^\omega}(n)\) to \(f_{\varepsilon_0}(n)\) *Linear array notation \(\{\underbrace{a,b\ldots y,z}_{n}\} \approx f_{\omega^\omega}(n)\) (limit) *Extended hyper-E notation \(xE\# \approx f_{\omega^\omega}(n)\) (limit) *n(k) function \(\approx f_{\omega^\omega}(n)\) *Taro's multivariable Ackermann function \(\approx f_{\omega^\omega}(n)\) *s(n) map \(\approx f_{\omega^\omega}(n)\) *Planar array notation \(\{a,b (2) 2\} \approx f_{\omega^{\omega^2}}(n)\) (limit) *Extended array notation (dimensional) \(\{a,b (0,1) 2\} \approx f_{\omega^{\omega^\omega}}(n)\) *BEAF superdimensional arrays \(\{a,b (\underbrace{0,0\ldots0,0,1}_{n}) 2\} \approx f_{\omega^{\omega^{\omega^\omega}}}(n)\) (limit) From \(f_{\varepsilon_0}(n)\) to \(f_{\Gamma_0}(n)\) Starting from here, the totality of these functions is not provable in Peano arithmetic. *BEAF tetrational arrays \({^ba} \& n \approx f_{\varepsilon_0}(n)\) (limit) *Cascading-E notation \(E\text{^} \approx f_{\varepsilon_0}(n)\) (limit) *m(n) map \(\approx f_{\varepsilon_0}(n)\) *Goodstein function \(G(n) \approx f_{\varepsilon_0}(n)\) *Hydra(n) function \(\approx f_{\varepsilon_0}(n)\) *X-Sequence Hyper-Exponential Notation \(\approx f_{\zeta_0}(n)\) *m(m,n) map \(\approx f_{\zeta_0}(n)\) *Nested Cascading-E Notation \(\approx f_{\varphi(\omega,0)}(n)\) From \(f_{\Gamma_0}(n)\) to \(f_{\omega^\text{CK}_1}(n)\) These functions and all those that follow cannot be proved total in arithmetical transfinite induction. *Extended Cascading-E Notation \(\approx f_{\vartheta(\Omega\omega)}(n)\) *tree(n) function \(\approx f_{\vartheta(\Omega^\omega)}(n)\) *TREE(n) function \(≥ f_{\vartheta(\Omega^\omega\omega)}(n)\) *BEAF (L-space) \(\approx f_{\vartheta(\Omega^\Omega)}(n)\) (according to Chris Bird) *Bird's H(n) function \(\approx f_{\vartheta(\varepsilon_{\Omega+1})}(n)\) *Bird's S(n) function (original) \(\approx f_{\vartheta(\theta_1(\Omega))}(n)\) *Bird's U(n) function \(\approx f_{\vartheta(\Omega_\omega)}(n)\) *SCG(n) function \(\geq f_{\psi_{\Omega_1}(\Omega_\omega)}(n)\) *BH(n) function \(\approx f_{\psi_0(\varepsilon_{\Omega_\omega + 1})}(n)\) *Bird's array notation \(\approx f_{\vartheta(\Omega_\Omega)}(n)\) (limit) *Bird's S(n) function (new definition) \(\approx f_{\vartheta(\Omega_\Omega)}(n)\) *Loader.c \(D(n) \ggg f_{\psi(\psi_{\alpha \mapsto I_{\alpha}}(0))}(n)\) *:No notation has yet been devised to describe the ordinal for this function. *Friedman's finite games *Friedman's finite trees Beyond \(f_{\omega^\text{CK}_1}(n)\) These functions are uncomputable, and cannot be evaluated by computer programs in finite time. *Rado's sigma function (and most of its variations) *''m''th order sigma function *Xi function *Rayo's function Other *Fusible margin function *: The growth rate of the \(m_1(x)\) function is an unsolved problem. *Laver tables *: It is not yet known if corresponding function is well-defined. *Hyperfactorial array notation *: HAN is new and still in development; its growth rate has not yet been conclusively evaluated. *Slow-growing hierarchy, Hardy hierarchy, Fast-growing hierarchy *: These three hierarchies can extend indefinitely, as long as ordinals and their fundamental hierarchies can be defined. *BEAF *: BEAF is not well-defined beyond tetrational arrays, therefore there are mutiple interpretations, and therefore there are also multiple growth rates possible. ja:関数の一覧 Category:Functions Category:Lists